Quantum phase synchronization via exciton-vibrational energy dissipation sustains long-lived coherence in photosynthetic antennas

The lifetime of electronic coherences found in photosynthetic antennas is known to be too short to match the energy transfer time, rendering the coherent energy transfer mechanism inactive. Exciton-vibrational coherence time in excitonic dimers which consist of two chromophores coupled by excitation transfer interaction, can however be much longer. Uncovering the mechanism for sustained coherences in a noisy biological environment is challenging, requiring the use of simpler model systems as proxies. Here, via two-dimensional electronic spectroscopy experiments, we present compelling evidence for longer exciton-vibrational coherence time in the allophycocyanin trimer, containing excitonic dimers, compared to isolated pigments. This is attributed to the quantum phase synchronization of the resonant vibrational collective modes of the dimer, where the anti-symmetric modes, coupled to excitonic states with fast dephasing, are dissipated. The decoupled symmetric counterparts are subject to slower energy dissipation. The resonant modes have a predicted nearly 50% reduction in the vibrational amplitudes, and almost zero amplitude in the corresponding dynamical Stokes shift spectrum compared to the isolated pigments. Our findings provide insights into the mechanisms for protecting coherences against the noisy environment.

Circular dichroism (CD) spectra of the trimer and monomer (Supplementary Fig. 7a) were measured at room temperature on a Chirascan-plus circular dichroism spectrometer (Applied Photophysics, U.K.) in a 1 mm pathlength quartz cuvette.The CD spectrum (red) of the monomer, which is without excitonic interaction, was subtracted from that (blue) of the trimer with a scaling factor of 2.6.The resulting lineshape in Supplementary Fig. 7b was fitted with two Gaussian peaks to determine the excitonic splitting.This excitonic splitting energy obtained (310 cm -1 ) is similar to that obtained in previous work 3 .Since the excitonic CD band splitting is about twice the electronic coupling strength (J), we determined that J ≈ 155 cm -1 .

Comparison of non-resonance Raman spectra for the rAPC trimer and monomer
The Raman spectra were recorded by a Labram HR evolution Raman spectrometer (Horiba, USA) at room temperature.All the samples were freeze-dried into powder to obtain a strong enough Raman signal from chromophores.The excited wavelength was 785 nm, which led to non-resonance excitation.Raman spectra of the trimer and monomer obtained after the removal of the background signal and smoothing the data via adjacent data point averaging are shown in Supplementary Fig. 8.For better comparison of Raman intensities, both spectra were normalized to the maximum intensity, which was observed at 1640 cm -1 .
In contrast to the significantly reduced coherent amplitude observed for all three vibrational modes (270, 660, and 805 cm -1 ) in the integrated coherence spectra of trimers, only the low-frequency mode, 270 cm -1 showed a remarkable decrease in the Raman spectrum; the relative intensities of the vibrational frequencies 660 and 805 cm -1 (805 cm -1 was slightly blue-shifted to 820 cm -1 in the powder) in the near-resonant region were nearly unchanged.The difference in the Raman intensity reveals that the decrease in coherent amplitude at 270 cm -1 is primarily due to the reduced Huang-Rhys factor of excitons resulting from the conformational change of the α84 pigments after trimerization.The low distinction between these two diagonal peaks is due to the large inhomogeneous broadenings at room temperature, as shown in the linear absorption spectrum of the rAPC trimer (Fig. 1b in the main text).Therefore, we performed additional 2DES experiments at 77K to reduce the effect of inhomogeneous broadening.The rAPC trimer was dissolved in a 2:1 glycerol: buffer solution, and placed in a 500-μm quartz cell.An optical density of 0.45 was measured at 650 nm at 77 K.As shown in Supplementary Fig. 11, two diagonal peaks and the cross peak below the diagonal which corresponds to the energy transfer process can be clearly observed at low temperatures.The absorption peaks above the diagonal are also observed which can be assigned to the transitions of the excitonic levels to the higher excited states.

Supplementary Note 5: Global analysis of the spectral evolution in 2DES
In the rAPC trimer, the spectral evolution over t2 at two typical excitation frequencies, 15300 and 16000 cm -1 , is shown in detail in Supplementary Fig. 12a and 12b.Upon excitation at 15300 cm -1 , which corresponds to the peak absorption of the lower exciton in the trimer, the time-dependent spectra are similar.In addition to the overall decay of the spectra, the maximal peak undergoes a slight red shift from 15220 cm -1 to 15180 cm -1 and a slight spectral broadening.
Global fitting from 40 fs to 95 ps revealed four lifetimes (130 fs, 1.6 ps, 57 ps, and 1.6 ns) in the spectral evolution.The last lifetime, 1.6 ns, which corresponds to the reported ground-state population recovery lifetime due to fluorescence and radiationless decay, is a fixed constant during global analysis 4 .The second and the third lifetime constants may reflect the energy relaxation to the electronic ground state because their DAS exhibit a spectral profile similar to that of the longest component.The 130-fs component mainly corresponds to solvation processes, including dynamical Stokes shift and spectral diffusion.These two processes often co-occur so that the spectral features are mixed intensively and cannot be distinguished from each other.In addition to the exponential decay of the excited-state population, the kinetic trace of the diagonal peak at (15300, 15300) cm -1 (right panel of Supplementary Fig. 12a) also exhibits a large-amplitude coherent (oscillation) signal within 1 ps.Supplementary Figure 12.Global analysis of spectral evolution at typical excitation frequencies for the trimer and the α-subunit.Spectral evolution and lifetime analysis at excitation frequencies of 15300 (a) and 16000 cm -1 (b) for the rAPC trimer and an excitation frequency of 16000 cm -1 (c) for the α-subunit.The spectra obtained at different waiting times are plotted in the left panels.Decay-associated spectra (DAS) with the corresponding lifetimes were obtained from a four-exponential fit of the sliced spectra from 40 fs to 95 ps in the middle panels.
In the right panels, the open circles represent the measured data points, and the fitted lines on the symbols were obtained by global fitting.The residual dynamics (purple lines) indicate the prominent oscillations within the first 1 ps.Source data are provided as a Source Data file.
Upon excitation at 16000 cm -1 , which corresponds to the upper exciton in trimer, the broad shoulder band decayed rapidly, within 400 fs, and nearly vanished after 2 ps.At longer waiting times, the amplitude of the main bleaching band decreased, and its spectral maximum shifted to the red slightly.Global fitting also revealed four lifetime constants (220 fs, 950 fs, 40 ps, and 1.6 ns) in the spectral evolution.DAS of the first two lifetime constants revealed the transfer of excitation energy from the upper exciton to the lower exciton in the trimer.The bi-exponential energy transfer process may arise from non-Gaussian microscopic heterogeneity in the distribution of energy transfer timescales, which was recently observed by the Schlau-Cohen group using single-molecule pump-probe spectroscopy 5 .Nonetheless, the first lifetime (220 fs) is closer to the timescale of energy transfer according to previous studies of the APC trimer. 6,7 the α-subunit, the spectral evolution over t2 at the excitation frequency 16000 cm -1 is shown in detail in Supplementary Fig. 12c.The spectral evolution of the α-subunit is analogous to that of the trimer excited at 15300 cm -1 (Supplementary Fig. 12a), and DAS of the first two lifetime constants (140 fs and 2.4 ps) show a spectral lineshape similar to that of the fastest lifetime (130 fs) in the trimer corresponding to both Stokes shift and spectral diffusion processes.
The third lifetime (26 ps) can be interpreted similarly to the third lifetime (57 ps) in Supplementary Fig. 12a as the energy relaxation to the ground state.

Supplementary Note 6: Kinetic analysis of the excited-state absorption signals
On the 2D spectra of rAPC, the peaks below the diagonal with negative amplitude are the excited-state absorption (ESA) peaks.For the monomer and two subunits, its central peak position locates at ω1=16400, ω3=14960 cm -1 ; while for the trimer, two peaks locate at ω1=16000, ω3=14760 cm -1 (ESA-1) from the upper excitonic state and ω1=15400, ω3=14600 cm -1 (ESA-2) from the lower excitonic state, respectively.One of the major differences in the 2D spectra of the trimer and the monomer is the absence of the ESA-1 signal after 400 fs.The absence results from the ultrafast energy transfer process between the upper and lower excitonic state.The kinetics of these ESA signal are shown in Supplementary Fig. 13a.Through a three-exponential function fit from 40 fs to 95 ps, we extracted the fastest lifetime constants for these three traces: 142 ± 61 fs (monomer), 187 ± 115 fs (ESA-1), and 111 ± 59 fs (ESA-2).The fast decay of ESA-1 is consistent with the energy transfer lifetime of 220 fs, while the faster relaxation of monomer and ESA-2 corresponds to solvation process as elucidated in Supplementary Fig. 12.
To consider the compensation effect with the negative ESA signal for the data analysis of the monomer and two subunits, we separated its contributions into two parts: 1) for the decay kinetics, the ESA and the ground-state bleaching (GB)/excited-state emission (SE) peaks are well separated; 2) for the dynamical Stokes shift, the ESA (negative) and SE (positive) peaks on 2D spectra are all red-shifted as shown in Supplementary Fig. 13b.The results in Supplementary Fig. 13

clear shows
To extract the dynamical Stokes shift from 2D spectra, we analyzed the time-dependent frequency shift of the maximum peak on the 2D spectra (t2, ω3) sliced along the selected excitation frequencies: 15300 cm -1 for the trimer and 16000 cm -1 for monomer and the two subunits, which correspond to the absorption maxima.The main peak includes the contribution from ground-state bleaching and the stimulated emission, and only the stimulated emission leads to a red shift in the frequency during the spectral evolution.The typical sliced 2D spectra and time-dependent peak frequency changes (blue lines) are shown in Supplementary Figs.14a and   14b.The frequency resolution of these spectra was improved by spline interpolation of the data.
During the first 300 fs, the main peak underwent a significant frequency redshift.Meanwhile, pronounced oscillations were superimposed on the dynamical Stokes shift kinetics.To include the influence of vibrational coherences on dynamical Stokes shift in the lifetime analysis, we combined a Gaussian function and a damped sine function to fit the dynamics 8,9 , as shown in Supplementary Fig. 14c.Fitting parameters associated with the vibrational coherences are presented in Supplementary Tables 1 and 2. The dynamical Stokes shift results from BBTA experiments are shown in Supplementary Fig. 14d.In both experiments, the coherence lifetime was longer for the trimer than for the monomer and two subunits, indicating that coherence has a protective effect in the trimer.

Supplementary
where 2 1 2 S = is the Huang-Rhys factor and Δ is the displacement of the corresponding nuclear coordinate from its equilibrium state.Assuming the Heitler-London approximation, the Hamiltonian of the dimer is given by 14   where T is the kinetic energy, g is the electronic common ground state, i q is the ground state dimensionless coordinates, and ii qq  = +  is the excited state dimensionless coordinates of the ith molecule.The electronic excitation results in a shift along the dimensionless coordinate q of the potential energy surface defined in the ground state as 2 () 2 The Schrödinger equation for the electronic states at the fixed nuclear configuration is solved as follows: ˆ( , ) ( , ) ( , ) ( , ) The corresponding secular equation is as follows: Thus, we get three eigen energies, which are the energy of the common ground state (E0) and the energies of the two electronic states (E±).
22 0 22 By assuming that the energy of the common ground state is zero (E0 = 0) and introducing the intermolecular symmetric collective coordinate 1 () 2 ==, we can simplify the eigen energies of the two electronic states (E±) as follows: Under the condition of weak electronic coupling J and a significant site energy difference  , the eigen energies can be further simplified as follows:

Exciton-vibration Hamiltonian of a dimeric model system with Jaynes Cummings form
Jaynes Cummings model can cast the Hamiltonian for wave functions within a local basis into that of a delocalized basis.The total Hamiltonian for wave functions of a local basis without the common ground state is is the delocalized new basis vector after diagonalizing.
Then, the final exciton-vibration Hamiltonian written in the Jaynes-Cummings form under the new basis vector is: ( ) By introducing the intermolecular symmetric collective coordinate 1 () 2 ( ) where the exciton-vibrational interaction is Eq. (2.6) clearly shows that only the anti-symmetric collective modes can be coupled to the two delocalized excitonic levels.
An approximate indicator describing the degree of coherent excitation transfer is derived from the transition probability between the two exciton-vibration states dominating electronic excitation transfer in the prototype dimer, which is an estimate of the maximum amplitude A for the population oscillations expressed as 15 where g is the corresponding vibrational strength.The equation shows that the resonant vibrational modes give rise to a maximum amplitude, and any detuning between the electronic splitting and the vibrational energy (  −  s ) would reduce the oscillation amplitude.On the other hand, a larger amount of electronic delocalization, i.e., a larger ( ) sin 2 , would also lead to a larger oscillation amplitude.

Dynamical Stokes shift in large polyatomic molecules in the condensed phase derived from a theory for femtosecond pump-probe spectroscopy
Based on the proposed theory for femtosecond pump-probe spectroscopy of large polyatomic molecules in the condensed phase, where a multimode Brownian oscillator model is used to account for high-frequency molecular vibrational and local intermolecular modes as collective solvent motion, Yan and Mukamel provided a semi-classical picture using the density matrix in Liouville space 16,17 .Within this frame, the pump field creates a doorway state that propagates for a specific time interval (waiting time), and the spectrum is calculated by finding its overlap with the window state.In particular, the doorway and window states are wave packets in the phase space.

Correlation function of the jth vibration mode within the Brownian oscillator under ultrafast pump-probe conditions
For a molecular model system with electronic-vibration interaction, the Hamiltonian can be expressed as follows: where j p is the dimensionless momentum, j q is the coordinate of the jth nuclear mode with frequency j  , and j d is the dimensionless displacement of the equilibrium configurations of this mode in the two electronic potential surfaces.j q may represent an intramolecular vibration, an intermolecular liberation, or a collective solvent motion.
Then assume that each mode experiences Brownian motion with a time-dependent Langevin friction function where the parameter =0  for the ground-state evolution and =1  for the excited-state evolution.
The correlation function of the jth mode within the Brownian oscillator model can be derived as follows: The frequency dependence of In this case by performing the inverse Laplace transform in Eq. (3.6), we get for the excited state, where p and * p are the momenta of the intramolecular mode and q and * q are the intramolecular coordinates for the ground and the excited states, respectively.In addition, we have where is the initial phase and A is an arbitrary constant.For the current experiment, in the high-frequency region (>500 cm -1 ), j    , and we have j j    .

Dynamical Stokes shift for the dimeric exciton in the condensed phase
We have shown that the upper and lower energy levels of the dimeric exciton can be expressed as where the superscript "*" refers to the excited-state properties.Eq 8 shows that the amount of dynamical Stokes shift depends on the phase relationship between the two intermolecular collective modes of the excited and ground states.In particular, when the motion of both

Figure 11 .
Absorptive 2D spectra of the rAPC trimer at 77 K. Dashed lines indicate the electronic transition energy (15300 and 15900 cm -1 ).Contour lines are drawn in 8.8 % intervals.The linear absorption spectrum (red) is shown on top of the 2D spectrum at 24 fs along with the laser spectrum (shaded area) used in the experiments.Source data are provided as a Source Data file.
we can simplify the exciton-vibration Hamiltonian as follows: stochastic random force due to solvent motion acting on the jth mode:

7 )
friction in the Laplace space, and s is the Laplace variable conjugate to time t: In this model, the time evolution of the jth mode depends entirely on its correlation function () j M t, which in turn depends on its harmonic frequency j  and the friction function ) (s j  .It can be found that ) (t M j satisfies the equation of motion: 0 scales of the thermal motions of the bath responsible for the random force.If these motions are very fast compared with the oscillator motion, the s dependence of ) (s j  is very weak and can be neglected, i.e., of the jth mode to dynamical Stokes shift will be thoroughly suppressed at the lower excitonic state.Supplementary References1 Liu, S. et al.Biosynthesis of fluorescent cyanobacterial allophycocyanin trimer in Escherichia coli.Photosynth.Res.105, 135-142 (2010).

Table 1 . Fitting parameters of the dynamical Stokes shift in the 2DES measurements of the rAPC trimer, monomer, α-and β-subunits.
The fitting function is d A and cA are the corresponding amplitudes,  ( ) is the oscillation frequency (phase), and C is the constant term..

Table 2 . Fitting parameters of the dynamical Stokes shift in the BBTA measurements of the rAPC trimer, monomer, α and β subunits.
The fitting function is same with the 2DES measurements.located above and below the diagonal in the rephasing coherence map, with a substantial enhancement of the ground-state vibration for both the trimer and α-subunit, where 660 and 805 cm -1 are in resonance with the electronic splitting, while 880 cm -1 may be in a quasi-resonance state.This clearly indicates that ground-state enhancement in the coherence map cannot be an indicator of the exciton-vibrational coupling.

Note 11: Detailed descriptions of the theoretical model 11.1 Dimeric model system in terms of collective vibrational coordinates
= , where subscripts A and B denote the two individual molecules in the dimer.The two monomers interact via a dipole-dipole coupling of strength J and are locally coupled to a quantized intramolecular

.2 Comparison of the Langevin equations for
When the solvation process is considered, the intra-molecular coordinate of the jth nuclear mode j q can represent local or collective solvent motion as well as intramolecular vibration and intermolecular liberation.In this case, we further assume that each mode experiences Brownian Q values.Compared with the upper excitonic level, the lower level has a longer lifetime and can be determined experimentally.Thus, the dynamical Stokes shift of the lower exciton level for jth mode can be expressed as: Substituting the underdamped nuclear motion of Eq. (3.16) into Eq.(3.18), finally, we have  